| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Multiple independent trials |
| Difficulty | Moderate -0.8 This is a straightforward application of independent probability with three events. Students multiply probabilities (using complements where needed) and add mutually exclusive outcomes. Part (a) and (b) are direct calculations, while (c) and (d) require combining cases but follow standard textbook patterns with no conceptual challenges. |
| Spec | 2.03a Mutually exclusive and independent events |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(D' \cap E' \cap F') = 0.4 \times 0.3 \times 0.2\) | M1 | At least 1 probability correct |
| \(= 0.024\) | A1 | CAO; OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(D' \cap E' \cap F) = 0.4 \times 0.3 \times 0.8\) | M1 | At least 2 probabilities correct |
| \(= 0.096\) | A1 | CAO; OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(\text{One}) = \text{(b)} + P(D \cap E' \cap F') + P(D' \cap E \cap F')\) | M1 | Use of 3 possibilities; ignore multipliers |
| \(= \text{(b)} + (0.6 \times 0.3 \times 0.2) + (0.4 \times 0.7 \times 0.2)\) | M1 | At least 1 new term correct |
| \(= 0.096 + 0.036 + 0.056 = 0.188\) | A1 | CAO; OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(\text{One or two}) = \text{(c)} + \text{(3 terms each of 3 probabilities)}\) or \(= 1 - \text{(a)} - \text{(1 term of 3 probabilities)}\) | M1 | \((c) + P(\text{Two})\); Used; OE; ignore multipliers; \(1-\text{(a)}-P(\text{Three})\) |
| \(= 0.188 + (0.6 \times 0.7 \times 0.2) + (0.6 \times 0.3 \times 0.8) + (0.4 \times 0.7 \times 0.8)\) \(= 0.188 + 0.084 + 0.144 + 0.224\) or \(= 1 - 0.024 - (0.6 \times 0.7 \times 0.8)\) \(= 1 - 0.024 - 0.336\) | M1 | At least 1 new term correct |
| \(= 0.64\) | A1 | CAO; OE |
# Question 5:
## Part 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(D' \cap E' \cap F') = 0.4 \times 0.3 \times 0.2$ | M1 | At least 1 probability correct |
| $= 0.024$ | A1 | CAO; OE |
## Part 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(D' \cap E' \cap F) = 0.4 \times 0.3 \times 0.8$ | M1 | At least 2 probabilities correct |
| $= 0.096$ | A1 | CAO; OE |
## Part 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(\text{One}) = \text{(b)} + P(D \cap E' \cap F') + P(D' \cap E \cap F')$ | M1 | Use of 3 possibilities; ignore multipliers |
| $= \text{(b)} + (0.6 \times 0.3 \times 0.2) + (0.4 \times 0.7 \times 0.2)$ | M1 | At least 1 new term correct |
| $= 0.096 + 0.036 + 0.056 = 0.188$ | A1 | CAO; OE |
## Part 5(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(\text{One or two}) = \text{(c)} + \text{(3 terms each of 3 probabilities)}$ or $= 1 - \text{(a)} - \text{(1 term of 3 probabilities)}$ | M1 | $(c) + P(\text{Two})$; Used; OE; ignore multipliers; $1-\text{(a)}-P(\text{Three})$ |
| $= 0.188 + (0.6 \times 0.7 \times 0.2) + (0.6 \times 0.3 \times 0.8) + (0.4 \times 0.7 \times 0.8)$ $= 0.188 + 0.084 + 0.144 + 0.224$ or $= 1 - 0.024 - (0.6 \times 0.7 \times 0.8)$ $= 1 - 0.024 - 0.336$ | M1 | At least 1 new term correct |
| $= 0.64$ | A1 | CAO; OE |
---5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are $0.6,0.7$ and 0.8 respectively.
Find the probability that, on a particular Sunday during the summer:
\begin{enumerate}[label=(\alph*)]
\item none of the three cyclists takes part;
\item Fabio is the only one of the three cyclists to take part;
\item exactly one of the three cyclists takes part;
\item either one or two of the three cyclists take part.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2007 Q5 [10]}}